The Fourier transform is used for characterizing linear systems when continuous waveforms are involved. However, for sampled data the discrete Fourier transform (DFT) is the tool that is applied. The fast Fourier transform is simply an efficient computational method for obtaining the DFT.
For a sequence Z.sub.0, . . . , Z.sub.N.sub.-1, of sampled waveform data (either real or complex), the DFT is defined as ##EQU1## For j = 0, 1, . . . , N- 1. (Some authors include a factor 1/N in the definition of W(j). An implicit assumption in the definition of W(j) is that the samples are obtained at equally spaced time intervals T.sub.0. The fundamental frequency is f.sub.0 = 1/T, where T = NT.sub.0, and the frequency corresponding to each index j is jf.sub.0. Several other properties of the DFT and FFT are described in "A Guided Tour of the Fast Fourier Transform," G. D. Bergland, IEEE Spectrum, Volume 6, p. 41-52, July 1969.